Nonparametric deconvolution of hormone time-series: A state-space approach *

Transcription

1 onparamerc deconvoluon of hormone me-seres: A sae-space approach * Guseppe De colao, Gancarlo Ferrar recae, Marco Franzos Dparmeno d Informaca e Ssemsca Unversà degl Sud d Pava Va Ferraa 7 Pava (Ialy el: Fax: E-mal: {dencolao,ferrar}@conpro.unpv. Absrac he nsananeous secreon rae (ISR of endocrne glands s no drecly measurable and can be reconsruced only ndrecly by applyng deconvoluon algorhms o me-seres of plasma hormone concenraons. In parcular, nonparamerc regularzaon-based deconvoluon hnges on a varaonal problem whose soluon s usually approxmaed by dscrezng he connuous-me axs. he paper shows how o perform regularzed deconvoluon avodng any form of dscrezaon. In vew of he equvalence beween regularzaon and Bayesan esmaon, s seen ha he esmaed ISR s a weghed sum of bass funcons, s he number of daa. Sae-space mehods are used o derve analycally he bass funcons as well as he enres of he marx of he lnear sysem used o compue he weghs. Alernavely, he weghs can be compued n O( operaons by a suable algorhm based on Kalman flerng. As an llusraon of he mehod, we esmae he sponaneous pulsale ISR of luenzng hormone (H from me seres of plasma H concenraons sampled every 5 mn. Keywords: Endocrnology, Hormones, Deconvoluon, Smoohng, Esmaon, Kalman flerng, neural newors.. Inroducon he nowledge of he glandular secreory acvy s essenal n order o undersand he complex feedforward/feedbac neracons of he endocrne sysem. However, he flux of hormone from he gland o he crculaon s no drecly measurable. I s only possble o oban me-seres of hormone concenraons measured n blood samples whdrawn a some gven nsans. In mos cases he hormone flux enerng he crculaon and he plasma hormone concenraon can be adequaely modeled as he npu and he oupu of a lnear me-nvaran sysem. hen, he ISR (nsananeous secreon rae of he gland can be raced bac by solvng a deconvoluon problem. o mae an example, consder he H (uenzng Hormone, whch plays an essenal role n reproducon physology and s secreed by he hypophyss. In order o assess s sponaneous secreon, a ypcal expermen nvolves collecng blood samples wh a 5 mn samplng perod over several hours (from 6 o 4. Snce H (as mos oher hormones s produced accordng o an epsodc pulsale paern, he possble parameers of neres are he number, he amplude, and he duraon of he ISR pulses [. he concenraon daa canno be used o drecly assess he gland acvy because he plasma H concenraon a a gven nsan s he resul of all he secreory acvy occurred before. he same nd of problems arse n he analyss of he sponaneous secreon of oher hormones such as FSH, ACH, Corsol, esoserone, ec, see e.g. [4, [5. here are some specfc feaures ha render he deconvoluon of hormone me-seres a nonrval as. Among hem, he ll-condonng of he nverse problem, he nfrequen and possbly nonunform samplng rae, he presence of nonnegavy consrans. In order o cope wh hese dffcules, several mehods have been proposed ncludng dscree deconvoluon, [3, parameer opmzaon [4, and, more recenly, nonparamerc regularzaon-based deconvoluon [6, [5, [. In he regularzaon approach, he npu s esmaed as he mnmzer of a cos funconal whch s he sum of he leas-squares f and a smoohness penaly, e.g. he negral * hs paper has been parally suppored by MURS proec"model Idenfcaon, Sysems Conrol, Sgnal Processng" and IH gran #RR95-.

2 of he squared frs (or second dervave of he npu. he acual compuaon s carred ou by dscrezng he unnown npu over a frequen grd (called "vrual" o dsngush from he samplng grd of he measuremens. Alhough, such an algorhm has been shown o perform effecvely n a number of cases [6, [, s only approxmaed and s accuracy depends on he choce of a suffcenly fne vrual grd. he man purpose of he presen paper s o develop explc formulas for he calculaon of nonparamerc regularzed deconvoluon avodng any form of dscrezaon. In fac, by explong he equvalence of regularzaon and Bayesan esmaon, s easy o see ha he soluon of he varaonal problem has he srucure of a regularzaon newor, [9.e. s a weghed sum of bass funcons. A frs conrbuon of he paper s he dervaon (by sae-space mehods of hese bass funcons when he mpulse response of he sysem s a lnear combnaon of exponenals (as happens for hormone me-seres. Sae space echnques play an mporan role also n he calculaon of he wegh vecor, whose calculaon as he soluon of a sysem of lnear equaons would requre O( 3 operaons. As a maer of fac, we show ha he weghs can be compued n O( operaons by means of a Kalman flerng algorhm. Fnally, o demonsrae he effecveness of he proposed mehod, we consder he deconvoluon of me-seres of plasma H concenraons wh 5-mn samplng perod.. Problem Saemen and Basc Assumpons Accordng o he leraure, see e.g. [4, [ and references quoed heren, a farly accurae model of he hormone concenraon dynamcs s gven by he convoluon negral z( g( τ u( τ dτ g( denoes he hormone elmnaon rae, and u( (miu/ml mn. s he hormone ISR, boh normalzed by he crculaon volume, whle z( (miu/ml s he hormone plasma concenraon. In oher words, g( s he mpulse response of he dynamcal sysem,.e. he hormone concenraon ha would be observed f, n absence of endogenous secreon, a unary nravenous hormone necon were made a me zero. u( I Sysem g*u z( z( v y ( Fg. : Bloc dagram of he deconvoluon problem; u(, unnown glandular secreon rae; z(: plasma hormone concenraon; y : dscree nosy measuremens. In he endocrnologcal leraure has been found ha a good model for g( s gven by a weghed sum of exponenals: ( g ( ae α Usually, 3. he parameers a and α depend on he specfc hormone (and possbly also on he pahophysologcal condons of he paen. A able summarzng he esmaed populaon parameers for several hormones (GH, ACH, PR, H, SH, Corsol, FSH s repored n [4. Dependng on he ype of expermen, a number of blood samples rangng from 5 o some hundreds are whdrawn. he samplng grd can be nonunform, wh samplng nervals rangng from mn. o 3 mn. Due o he measuremen errors he acual samples y (miu/ml are gven by y z( v,,,..., (3 he errors v are zero-mean, ndependen and normally dsrbued, wh Var[v >,. In general, alhough from a compuaonal pon of vew suffces ha >> (compared o he sysem me consans. From he saemen of he problem, s apparen ha he ISR esmaon problem amouns o reconsrucng he npu of a lnear sysem, gven nosy samples of he oupu. In he leraure, a number of deconvoluon mehods have been proposed, see [6 for a dscusson. In parcular, he regularzaon mehod [ s based on he cos funconal J ( u ( y z Var[ v ( du( τ dτ dτ he posve real number s he so-called regularzaon parameer whch conrols he rade-off beween fdely o he daa and smoohness of he soluon. Among he possble crera for he unng of hs parameer one may ce GCV (Generalzed Cross Valdaon [6 and M (Maxmum elhood [8, [6. he regularzed esmae s defned as he soluon of he varaonal problem ( (4 u$ ( arg mn J ( u (5 u( he sandard way o compue u$ ( s o dscreze he unnown funcon u( and reformulae (4, (5 as a quadrac opmzaon problem he unnown s a vecor, as he convoluon negral ( s approxmaed by a marx-vecor produc. hen he soluon s found by solvng a sysem of lnear equaons, wh complexy O( 3, see e.g. [6. We end hs secon by ponng ou a Bayesan nerpreaon of he esmaor (5, whch wll be useful n he followng. Hereafer, he vecor of expermenal daa wll be denoed by Y [ y y... y. he followng proposon s a corollary of heorem.5.. n [6. Proposon : Assume ha (: u ( w ( τ d τ, w( s a connuous-me WG (Whe Gaussan ose, ndependen on v, wh nensy λ ;

3 (: λ. hen, he regularzed soluon u$ ( concdes wh he condonal expecaon (Bayes esmae u$ ( E u( Y E u( Y E YY Y. (6 3. A Sae-Space Deconvoluon Formula In hs secon wll be shown ha he parcular srucure ( of he mpulse response g( can be exploed n order o derve an effcen compuaonal algorhm ha compleely avods dscrezaon. o hs purpose, consder he followng sochasc sae-space represenaon: x& Ax Bw y Cx( v x ( R, x ( ~ (, A, B, C are bloc marces defned as follows... α A, α G F, α B C [ H F dag{ } (8,,,, G [..., H [ a a... a. Under he assumpons of Proposon, s easy o see ha u( x ( and z( Cx(. Gven he sae-space represenaon (7, (8 a possble way o esmae u( s o resor o opmal sae-space smoohng algorhms [. Heren, however, we follow a dfferen approach ha leads o a smple analyc form for he esmae. Recallng ha u ( x (, he followng resul s a sraghforward consequence of Proposon. Proposon : Under he assumpons of Proposon, u$ ( ϑφ( (9 φ ( E[ x ( y ϑ [ ϑ ϑ ϑ (7 and he vecor... sasfes he equaon EYY [ ϑ Y (! Observe ha he esmae (9 has he srucure of a socalled regularzaon neural newor [9, φ (,,..., are he bass funcons wh correspondng weghs ϑ,,...,. An advanage of he newor (9 s ha, f dfferen expermens are carred ou usng he same samplng grd, he funcons φ ( and he srucure of he marx EYY reman unchanged. hen, each connuous-me esmae s compleely characerzed by s specfc -dmensonal wegh vecor ϑ. In he followng, wo man resuls wll be derved: ( an analycal expresson for φ ( and EYY wll be wored ou; ( wll be shown how o compue θ n O(operaons (nsead of O( 3 In order o fulfll he frs obecve, wo prelmnary lemmas are needed. emma : e A be as n (8. hen A e F, (.a ~ G ( e ~ ~ ~ ~ G ( [ G ( G (... G ( ~ e G ( (.b α F e dag{ e } emma : e X( E[x( x(. hen ~ x ( X( ~, ~ λ x R, x R ( x ( x ( ( ~ e x (, α α ( α α ( e x ( αα α α ( ( e e. α α Proof : I s well nown ha he sae covarance marx X(,, s he soluon of he yapunov dfferenal equaon X & ( AX ( X ( A BB λ wh nal condon X(. he explc soluon s A ( τ A ( τ X( λ e BB e dτ λ Aτ A τ e BB e dτ. (3 hen, n vew of emma, he hess sraghforwardly follows by smple compuaons. heorem : Wh reference o Proposon, he bass funcons φ (,,,..., are gven by λ x~ ( H, > ~ (4 φ ( λ (( G( H F x ~ ( ( e H, Proof: Frs, observe ha

4 A ( τ e X( τ, τ Exx [ ( ( τ A ( τ Xe (, τ hen, f, we have ha φ Eu ( y Ex( z ( E x ( x ( C [ [, A ( [, e X( C. Conversely, f, φ A ( [, X ( e C hen, he hess follows n vew of emmas and. heorem : Wh reference o Proposon, he marx E[Y( Y( s gven by: E[Y Y Σ dag{ Var[v }, Σ Ez [ ( z ( λ [ F( ~ HG ( ~ (5 x ( e x ( H, Σ, < Proof : If, we have ha [ CE[ A ( Ez ( z ( x ( x ( C Ce X ( C Analogously, f, A ( Ez ( z ( CX ( e C hen he hess follows by applyng emmas and. 4. Wegh Calculaon va Kalman Flerng In general, he calculaon of he wegh vecor ϑ hrough he soluon of he sysem of lnear equaons ( would requre O( 3 operaons. In hs secon an effcen O( algorhm based on a Kalman fler s derved. o hs purpose, a defnon and a echncal lemma are frs nroduced. Defnon: In he followng, he lnear me-varyng dscree-me sysem x Ax Bu x (6.a y Cx Du,,...,, (6.b u, y R, s sad o be a realzaon of he marx M R f y y... y M u u... u. emma 3: Assume ha (6 s a realzaon of M. hen, he sysem ξ Aξ Cυ ξ η Bξ Dυ, -,..., s a realzaon of M, n he sense ha η η... η M υ υ... υ Proof: By nspecon. heorem 3: Consder he sochasc sysem (7, and le ϑ [ ϑ ϑ... ϑ be a soluon of EYY ϑ Y. hen, ϑ can be compued by means of he followng O( algorhm: A ξ ( ( e KC ξ Ky, ξ ( ξ (7.a η R - y C,,..., (7.b Ψ ( ( A e KC Ψ C R - η, Ψ (8.a - ϑ KΨ R η,, -,..., (8.b A ( P e A Pe ( Q KRK, P ( τ ( τ τ ( A A Q λ e BB e d X A ( R CPC Var v, K e PC R Proof: oe ha (7.a s us he -sep Kalman predcon for he dscree-me sysem obaned by samplng he connuous-me sochasc sysem (7. In parcular, { η } s he sequence of he normalzed nnovaons. By a well-nown propery of Kalman flerng heory [, he nnovaons sequence s whe so ha, leng η [ η η... η E ηη I. ow, follows ha le Γ be he marx such ha η Γ Y (n oher words, (7 s a realzaon of Γ. hen I E ηη E ΓY Y Γ ΓE Y Y Γ whch easly mples E [ ηη Γ Γ. Hence, leng ϑ be defned by (7-(8, we conclude ha ϑ Γ ΓY E YY Y. 5. Deconvoluon of H daa he algorhm wored ou n he prevous secons has been esed on a me-seres of plasma H (luenzng hormone concenraons n a normal subec conssng of he samples y,,,4, colleced wh a unform 5- mn samplng perod ( 5( - [7. o ease he comparson wh he dscrezaon approach o nonparamerc deconvoluon, we analyze he same se of daa presened n [6. he mpulse response descrbng he hormone decay n he crculaon s gven by he second-order model (: g( a e - α a e - α a, a, α, α are populaon values ( a.65 miu ml -, a.385 miu ml -, α mn -, α mn -, miu sands for mll-inernaonal-un, see [3. he meas-

5 uremen error has a consan Coeffcen of Varaon, namely Var[ v σ y, σ 5.% s he coeffcen of varaon. Snce sponaneous secreon s suded, u(, <, whch has been ep no accoun by leng - 55 mn. he regularzaon parameer was adused so as o oban he same degree of smoohness as n a prevous sudy [6. hs was acheved by unng unl he equvalen degree of freedom q( was equal o 6.6 (for he defnon and a dscusson of he role of q( n he opmal unng of, he neresed reader s referred o [6. he resuls are repored n Fg.. In he upper panel, he measured concenraons are ploed ogeher wh he connuous-me concenraon profle reconsruced by reconvolung he esmaed ISR, whch n urn s repored n he lower panel. he pulsale naure of he ISR s apparen. In parcular s easy o dsngush 4-5 maor secreory epsodes as well as some mnor ones. A comparson wh Fg. n [6 does no show any apprecable dfference wh respec o he esmaes obaned by dscrezaon. everheless, such a dscrezaon had been carred ou on a raher frequen vrual grd (-mn samplng correspondng o a 5 unnown vecor whch enals a sgnfcan compuaonal burden. On he conrary, he new deconvoluon procedure does no use any dscrezaon and s compuaonally more effcen. Concenraon [ miu / ml ISR [ miu / ml mn Reconvoluon of H hormone Esmaed ISR of H hormone 5 5 me [mn Fg. : Deconvoluon of H daa. Upper panel: measured H concenraons n plasma (crcles and reconsruced connuous-me concenraon profle (-. ower panel: esmaed ISR. 6. References [ Anderson, B.D.O. and J.B. Moore (979. Opmal Flerng. Prence-Hall, Englewood Clffs, J. [ Berman, G.J. (973. Fxed nerval smoohng wh dscree measuremens. In. J. Conrol, 7, [3 Cobell, C., A. Mar, S. Del Prao, S. De Kreuzenberg, R. osadn and I. Jensen (987. Reconsrucng he rae of appearance of subcuaneous nsuln by deconvoluon. Am. J. Physol., 5, E549-E556. [4 De colao, G. and D. bera (993. near and nonlnear echnques for he deconvoluon of hormone me-seres. IEEE rans. on Bomed. Eng., 4, [5 De colao, G., D. bera and A. Saroro (995. Deconvoluon of nfrequenly sampled daa for he esmaon of growh hormone secreon. IEEE rans. on Bomed. Eng., 4, [6 De colao, G., G. Sparacno and C. Cobell (997. onparamerc npu esmaon n physologcal sysems: Problems, mehods and case sudes. Auomaca, 33, [7 Genazzan, A.D., G. For, M. Magg, M. Mllon, F. Canfanell, V. Guardabasso, V. oscano, M. Sero and D. Rodbard (99. Pulsale secreon of luenzng hormone n agonadal men before and durng esoserone replacemen herapy. J. Endocrnol. Inves., 3, [8 MacKay, D.J.C. (99. Bayesan Inerpolaon. eural Comp., 4, [9 Poggo,., and F. Gros (99. ewors for approxmaon and learnng. Proc. IEEE, [ Saroro, A., G. De colao, G. Pzzn and D. bera (997. on-paramerc deconvoluon provdes an obecve assessmen of GH responsveness o GH-releasng smul n normal subecs. Cln. Endocrnol., 46, [ honov, A.., V.Y. Arsenn (977. Soluons of Ill-Posed Problems. Wnson/Wley, Washngon. [ Urban, R.J., W.S. Evans, A.D. Rogol, D.. Kaser, M.. Johnson, and J.D. Veldhus (988. Conemporary aspecs of dscree pea-deecon algorhms. I. he paradgm of he uenzng Hormone pulse sgnal n men. Endocrne Revews, 9, [3 Veldhus, J.D., F. Fraol, A.D. Rogol, and M.. Dufau (986. Meabolc clearance of bologcally acve luenzng hormone n man. J. Cln. Inves., 77, -8. [4 Veldhus, J.D., M.. Carlson & M.. Johnson (987 he puary gland secrees n burss: Apprasng he naure of glandular secreory mpulses by smulaneous mulple-parameer deconvoluon of plasma hormone concenraons. Proceedngs of he aonal Academy of Scences of he Uned Saes of Amerca, 4, [5 Veldhus, J.D., & M.. Johnson (995 Evoluon of deconvoluon analyss as hormone pulse deecon mehod. In: Quanave euroendocrnology (M.. Johnson and J.D. Veldhus, eds., Mehods n euroscences Seres, Vol. 8, Academc Press, San Dego, pp. -4. [6 Wahba, G. (99. Splne Models for Observaonal Daa SIAM, Phladelhpa.